Mathematics Colloquia Archives - Spring 2014

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​Infinite Dimensional Lie Algebras and Modular Forms

Dr. K yu-Hwan Lee (University of Connecticut)

February 6, 2014

Abstract: In this talk, we will observe how modular forms naturally arise from the theory of infinite dimensional Lie algebras. In particular, we will consider affine and rank 2 hyperbolic Kac--Moody algebras and see their relationship to Jacobi modular forms and Hilbert modular forms, respectively.

Using Learning Catalytics to Create an Interactive Classroom

Abstract: Peer instruction and other interactive teaching methods have been shown to dramatically improve conceptual understanding. While no technology is necessary to take advantage of these teaching methods, technology can enable the instructor to better understand student understanding, differentiate instruction, and facilitate more productive student discussions in the classroom. This workshop will introduce Learning Catalytics, a cloud-based platform for interactive teaching that allows students to use web-enabled devices—laptops, smartphones, and tablets—to engage in rich, authentic tasks in class and allows instructors to go beyond clickers and other response systems to create an interactive environment that integrates assessment with learning. This presentation is sponsored by Pearson Education.​

Primes, Zeros, and Random Matrix Theory

Professor Hugh Montgomery (University of Michigan)

March 6, 2014

Abstract: In 1859, Riemann showed that the distribution of prime numbers depends on the location of the zeros of the Riemann zeta function. Hilbert and Pólya independently speculated that the Riemann Hypothesis (RH) is true because the zeros are related to some (still undiscovered) Hermitian operator. For forty years we have had some limited evidence that the zeros are spectral. Today, Random Matrix Theory is a valuable tool that enables us to generate deep conjectures concerning the zeta function.

Isospectral Surfaces of Small Volume

Dr. Benjamin Linowitz (University of Michigan)

March 25, 2014

Abstract: In 1980 Vigneras used orders in quaternion algebras to exhibit pairs of hyperbolic 2-manifolds which were isospectral but not isometric, thereby proving that one cannot hear the shape of a 'hyperbolic drum'. The example which appeared in Vigneras' paper turned out to be incorrect for a technical reason, and was subsequently corrected in her book on quaternion orders. The corrected manifolds turn out to have enormous genus: 100801. In this talk we will exhibit substantially simpler examples; for instance, a pair of isospectral hyperbolic 2-orbifolds whose underlying surfaces have genus 0. These examples are further distinguished in that they have minimal volume amongst those arising from maximal arithmetic Fuchsian groups and cannot be obtained via Sunada's method. This is joint work with John Voight.

The Curse of Dimensionality in a Real Life Industrial Problem

Abstract: Industrial data mining (supervised learning) problems generally involve wrestling with the "curse of dimensionality". Data collected in an industrial transaction environment is rarely if ever intended to be used in a modeling problem, let alone in a "cause and effect" modeling problem. Thus the contradiction between cause and effect (ala the use of the scientific method and proper design of experiments) and "prediction" is before us. This curse of dimensionality seems to prevent analytics professionals from finding true cause and effect. Data sets not intended for modeling generally have significant multicollinearity, lack of balance and often are too wide (p being inappropriate for n). Approaches for solving these issues can be broken down in three classes; dimension reduction, parameter adjustment and data structure adjustment. This talk will show an industrial data mining problem where the curse is present in all its glory. Each of the three basic methods for supposed "solutions" to the problem will be presented using modern day technologies.

Geometric Properties of 2-Step Nilpotent Lie Groups

Dr. Rachelle DeCoste (Wheaton College)

April 4, 2014

Abstract: 2-step nilpotent Lie groups with a left invariant metric are objects that admit both positive and negative sectional and Ricci curvatures. The geometry of such an object reflects strongly the algebraic structure of the associated Lie algebra. We will discuss various geometric properties of Lie groups such as the presence of closed geodesics. We will also define some classes of 2-step nilpotent Lie groups, namely Heisenberg type and Heisenberg-like, and give examples of each. Finally, new and ongoing research into the properties of 2-step nilpotent Lie groups associated to simple graphs will be introduced.​

Examples of Singular Perturbations in Variational Problems

Professor Aaron Yip (Purdue University)

April 10, 2014

Abstract: We will discuss the idea of Gamma convergence which is used to identify limiting description of singularly perturbed problems in variational settings. The prototype is the Allen-Cahn Functional which is used in the modeling of phase transitions and phase boundary motions. It can also be connected to minimal surfaces and motion by mean curvature. Examples will be given to demonstrate the versatility of this framework. On the other hand, I will also give some examples that might not be easily handled by this approach.

Symbolic Powers of Ideals: Containments and Conjectures

Dr. Mark Johnson (University of Arkansas)

April 15, 2014

Abstract: The symbolic power of an ideal, in a commutative noetherian ring, is the "pure" part of the ordinary power of the ideal - in the sense of the Lasker-Noether primary decomposition. Symbolic powers occur frequently in commutative algebra and algebraic geometry and there are many open problems about them, roughly concerning their growth - how much larger they are compared to the ordinary power. In this talk, we will give a survey of recent progress on a number of these problems regarding them.

On Applications of Suslin's Local Global Principle

Abstract: In 1975 Daniel Quillen and Andrei Suslin independently proved the problem posed by J-P. Serre in 1955 on finitely generated projective modules, which asks whether finitely generated projective modules over a polynomial ring over a field are free. In this talk we shall concentrate on A. Suslin's local-global principle which is one of the main ingredient for Suslin's proof of Serre's conjecture. We shall discuss various applications of this theorem in the field of classical K-theory. ​